A person is holding on to a helium balloon in an elevator. The elevator cable snaps, and at the exact same moment, the person lets go of the helium balloon. In the perspective of the person in the elevator, what happens to the balloon the moment the cable snaps?

Buoyancy is a direct consequence of gravity. In free fall, the effective gravity within the elevator is zero (assuming perfect free fall of the elevator, with no air drag) and thus the balloon stays where it is relative to the observer.

Comments

When the lift is in free-fall, wouldn't it become weightless and therefore its contents would also become weightless? I would think that this would make the balloon drop towards the floor of the elevator. What's your reasoning Timmmm?

The fundamental reason why the helium filled balloon floats is buoyancy. Buoyancy is a by-product of acceleration from the natural state of falling with gravity. In this case, he floor of the elevator is accelerating at exactly +g to stop you from descending. In free fall, the effective acceleration within the elevator is reduced to zero (or very close to it) and buoyancy no longer plays a factor. Thus the balloon stays where it is. This is the same fundamental reason why astronauts on the ISS float.

In the confines of an elevator shaft where the area of the base of the elevator covers most of the cross-sectional area of the shaft, I would imagine the elevator would reach terminal velocity quite quickly and stop accelerating at G, at which point the balloon would go up. So I guess your answer is assuming an ideal case where the lift keeps accelerating at G?